Finite Group Databasebeta

← #11786012

Subgroups — #11786012

Order 1152.

409 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10592696C211yesmaximal, characteristic
2C2×#63578C211yesmaximal, characteristic
3#10591138C211yesmaximal, characteristic
4C3×#4531C212yesmaximal
5#10589170C212yesmaximal
6#68699—32nomaximal
7C32×#117C2211yescharacteristic
8C2×#3748C2211yescharacteristic
9C22×C3⋊(C3⋊C8)C2211yescharacteristic
10C3×#926C2212yes—
11C2×C3×#208C2212yes—
12#63630C412yescomplemented
13C4×C3⋊(C3⋊C8)C412yescomplemented
14#63578—21no—
15#63578—21no—
16#63576—21no—
17#63574—21no—
18C2×#822—32no—
19#4998—32no—
20#4997—32no—
21#4997—61no—
22C3×H(3,ℤ/4)—32no—
23#4559—32no—
24#4533—32no—
25#4531D312yes—
26C22×C32×C4C2311yescharacteristic
27C32×C4⋊C4—21no—
28C32×C22⋊C4—21no—
29C32×C42C2×C412yes—
30#3748—21no—
31#3748—21no—
32C2×C3⋊(C3⋊C8)C2×C412yes—
33C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
34C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
35C2×C3⋊(C3⋊C8)D411yescharacteristic
36C2×C3⋊(C3⋊C8)D411yescharacteristic
37C2×C3⋊(C3⋊C8)Q811yescharacteristic
38C2×C3⋊(C3⋊C8)Q811yescharacteristic
39C3×#217—22no—
40C3×C4⋊Dic3—22no—
41C3×C4×Dic3—22no—
42C3×#208—22no—
43C3×#208—22no—
44C2×C3×C3⋊C8—22no—
45#1833—91noSylow
46C2×C3×M16—32no—
47C3×#117—21no—
48C3×#117—32no—
49C3×#117C2×D312yes—
50#926C2×D312yes—
51C2×#208—32no—
52C2×#208—61no—
53C2×#208C2×D312yes—
54C22×C3⋊C8—32no—
55C22×C3⋊C8—61no—
56#822—62no—
57#822—62no—
58#808—62no—
59#807—62no—
60#803—34no—
61#803—62no—
62C4×C3⋊C8—34no—
63C4×C3⋊C8—62no—
64C23×C32#4111yescharacteristic
65C2×C32×C4—22no—
66C2×C32×C4C22×C411yescharacteristic
67C2×C32×C4C2×D411yescharacteristic
68C2×C32×C4C2×Q811yescharacteristic
69C2×C32×C4#4111yescharacteristic
70C2×C32×C4#4111yescharacteristic
71C2×C32×C4#4111yescharacteristic
72C2×C3×Dic3—42no—
73C3⋊(C3⋊C8)—22no—
74C3⋊(C3⋊C8)C4⋊C411yescharacteristic
75C3⋊(C3⋊C8)C4⋊C411yescharacteristic
76C3⋊(C3⋊C8)C4⋊C411yescharacteristic
77C3⋊(C3⋊C8)C4⋊C411yescharacteristic
78C3×C3⋊C8—42no—
79C2×#108—91no—
80#427—91no—
81#405—91no—
82#344—92no—
83H(3,ℤ/4)—32no—
84C22×C3×C4—21no—
85C22×C3×C4C22×D312yes—
86C3×M16—62no—
87C3×M16—62no—
88C2×C3×C8—62no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—62no—
92C3×C22⋊C4—22no—
93C3×C22⋊C4—22no—
94C3×C22⋊C4—62no—
95C3×C42—22no—
96C3×C42—62no—
97C3×C42C4×D314yes—
98#217—22no—
99C4⋊Dic3—22no—
100C4×Dic3—22no—
101#208—22no—
102#208—22no—
103#208—62no—
104#208—62no—
105#208—62no—
106#208—62no—
107C2×C3⋊C8—22no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—32no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—32no—
112C2×C3⋊C8—32no—
113C2×C3⋊C8—32no—
114C2×C3⋊C8—34no—
115C2×C3⋊C8—61no—
116C2×C3⋊C8—62no—
117C2×C3⋊C8—61no—
118C2×C3⋊C8—61no—
119C2×C3⋊C8—61no—
120C2×C3⋊C8—61no—
121C2×C3⋊C8—61no—
122C22×C32—41no—
123C22×C32#11711yescharacteristic
124C22×C32#12411yescharacteristic
125C22×C32#12511yescharacteristic
126C32×C4—42no—
127C32×C4C2×C4⋊C411yescharacteristic
128C32×C4#11711yescharacteristic
129C32×C4#12411yescharacteristic
130C32×C4#12511yescharacteristic
131C3×Dic3—82no—
132C2×M16—32no—
133C2×M16—91no—
134C22×C8—91no—
135#117—32no—
136#117D3211yescharacteristic
137#108—181no—
138#108—181no—
139#107—181no—
140#106—181no—
141#97—92no—
142C4×C8—92no—
143C23×C3—21no—
144C23×C3#23712yes—
145C2×C3×C4—24no—
146C2×C3×C4—24no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—122no—
154C2×C3×C4C2×C4×D312yes—
155C2×C3×C4#23512yes—
156C2×C3×C4#23512yes—
157C2×C3×C4D3×D412yes—
158C2×C3×C4#23712yes—
159C2×C3×C4D3×Q812yes—
160C2×Dic3—42no—
161C3×C8—122no—
162C3⋊C8—32no—
163C3⋊C8—32no—
164C3⋊C8—32no—
165C3⋊C8—32no—
166C3⋊C8—42no—
167C3⋊C8—61no—
168C3⋊C8—61no—
169C3⋊C8—61no—
170C3⋊C8—61no—
171C3⋊C8—64no—
172C3⋊C8—64no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32#38211yescharacteristic
177C22×C4C2×D3211yescharacteristic
178M16—62no—
179M16—62no—
180M16—181no—
181M16—181no—
182C2×C8—62no—
183C2×C8—91no—
184C2×C8—91no—
185C2×C8—91no—
186C2×C8—91no—
187C2×C8—91no—
188C2×C8—91no—
189C2×C8—92no—
190C4⋊C4—21no—
191C4⋊C4—62no—
192C22⋊C4—21no—
193C22⋊C4—62no—
194C42—62no—
195C42#62412yes—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3#87712yes—
202C22×C3#88512yes—
203C22×C3#88912yes—
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—21no—
207C3×C4—21no—
208C3×C4—44no—
209C3×C4—44no—
210C3×C4—242no—
211C3×C4#87212yes—
212C3×C4#88512yes—
213C3×C4#88912yes—
214C3×C4D3×C4⋊C412yes—
215Dic3—82no—
216C32#183311yescharacteristic, complemented, Sylow
217C23#380511yescharacteristic
218C2×C4—22no—
219C2×C4—122no—
220C2×C4#379711yescharacteristic
221C2×C4#379811yescharacteristic
222C2×C4#379811yescharacteristic
223C2×C4#380211yescharacteristic
224C2×C4#380511yescharacteristic
225C2×C4C2×#62411yescharacteristic
226C8—91no—
227C8—91no—
228C8—91no—
229C8—91no—
230C8—122no—
231C8—182no—
232C2×C3—22no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—22no—
236C2×C3—21no—
237C2×C3—41no—
238C2×C3—41no—
239C2×C3#465012yes—
240C22—41no—
241C22#6387011yescharacteristic
242C22#6391511yescharacteristic
243C22#6395711yescharacteristic
244C4—42no—
245C4—242no—
246C4#6387011yescharacteristic
247C4#6387411yescharacteristic
248C4#6390011yescharacteristic
249C4#6391511yescharacteristic
250C3—21no—
251C3#6869912yescomplemented
252C2—21no—
253C2—21no—
254C2—21no—
255C2#1059136011yescharacteristic
subgroupG/H
centerC4#63874
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#117C22
socleC2×C32#382
prime radicalC23×C32#41
cosocleC22×C32×C4C23
hypercenter#117D32
hypocenterC32#1833
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ #117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. #63630C4semidirect
13. C4×C3⋊(C3⋊C8)C4semidirect
216. C32#1833semidirect
251. C3#68699semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#105926962
C2×#635782
#105911382
C3×#45312
#105891702
#686993